Number Of Solutions To A Class Of Equations Over Finite Fields

In number theory, finite field is not only basic but also important, especially equations over finite fields.Nowadays, equations over finite fields are deeply studied by more and more scholars, particularly the numberof solutions of equations over finite fields. Until now, we can get the number of solutions of some equationsover finite fields. But in most cases, it is difficult to get an accurate value and we only get the estimate one(unless some additional conditions). This paper aims to solve the problem about the number of solutions of aclass of equations over finite fields, and obtain the formula of the number of solutions of the equations undercertain limited conditions. The first part is the introduction. The second part mainly introduces the basicknowledge related to our paper, including groups, rings, fields, finite fields, effective reduction, Elliptic Curvesand etc. The estimation of the number of zeros of polynomials and the formula of the number of solutions ofsome equations are discussed in the third part. The fourth part is the core content of this paper. Through avariety of conversion methods (such as effective reduction, combination and so on), we get the formula of thenumber of solutions of a class of equations over finite field.The last part is the summary.